The low-dimensional homotopy of the stable mapping class group

TL;DR

This paper investigates the integral cohomology of the stable mapping class group, computes its first four cohomology groups, and explores the Postnikov tower of its classifying space, revealing new geometric and algebraic insights.

Contribution

It provides the first detailed analysis of the integral cohomology of the stable mapping class group and computes the initial stages of its Postnikov tower, using novel geometric and algebraic methods.

Findings

Computed the first four integral cohomology groups of the stable mapping class group.

Described an explicit geometric generator of the third homotopy group.

Analyzed the Postnikov tower of the classifying space of the stable mapping class group.

Abstract

Due to the deep work of Tillmann, Madsen, Weiss and Galatius, the cohomology of the stable mapping class group $\gaminf$ is known with rational or finite field coefficients. Little is known about the integral cohomology. In this paper, we study the first four cohomology groups. Also, we compute the first few steps of the Postnikov tower of $B \gaminf^+$, the Quillen plus construction applied to $B \gaminf$. Our method relies on the Madsen-Weiss theorem, a few known computations of stable homotopy groups of spheres and projective spaces and on a certain action of the binary icosahedral group on a surface. Using the latter, we can also describe an explicit geometric generator of the third homotopy group $\pi_3 (B \gaminf)$.